Give example of a countable collection of finite sets whose union is not finite

This is exercise 3.E from Elements of Real Analysis by Bartle. The exercise asks to give an example of a countable collection of finite sets whose union is not finite and I wanted to verify if my solution is correct or not. My proposed solution is:

Let $X_n = \{1, 2, \ldots, n\}$ for some $n \in \mathbb{N}$ and let $\mathcal{X} = \{X_1, X_2, X_3, \ldots\}$. Then each $X_i$ in $\mathcal{X}$ is finite, $\mathcal{X}$ is countable, and $\bigcup \mathcal{X}$ is not finite.